Find one value of $x$ that is a solution to the equation: $(7x+2)^2+6(7x+2)=27$ $x=$
Answer: We could solve for $x$ by expanding $(7x+2)^2$, combining terms that are alike, and using the quadratic formula or factoring to solve for $x$. However there is a more elegant way to approach this problem. Let's use structural features to rewrite the equation in a simpler form. Let's look at the given equation: $({7x+2})^2+6({7x+2})=27$ If we let ${p}={7x+2}$, we can see that this equation is in the form: ${p}^2+6{p}=27$ Let's solve this equation in terms of ${p}$ : $\begin{aligned}{p}^2+6{p}&=27\\\\ {p}^2+6{p}-27&=0\\\\ ({p}+9)({p}-3)&=0\\\\ {p}=-9\ &\text{or} \ \ {p}=3 \end{aligned}$ Since ${p}={7x+2}$, let's substitute this value back into our two solutions in order to solve for $x$ : ${7x+2}=-9\ \ \ \text{or} \ \ \ {7x+2}=3$ When we solve ${7x+2}=-9$, we find that $x=-\dfrac{11}{7}$. When we solve ${7x+2}=3$, we find that $x=\dfrac{1}{7}$. In conclusion, the two solutions of the equation $(7x+2)^2+6(7x+2)=27$ are $x=-\dfrac{11}{7}$ and $x=\dfrac{1}{7}$. [Is there another way to solve for x?]